Christine Quinn gave this city a lesson about our emergency services, even if it was not the one she intended.

At an outdoor event in Brooklyn where Quinn was speaking on Tuesday, a City Council intern fainted. 911 was called, and when an ambulance did not appear, Quinn called both the fire and police commissioners to get them to speed it up. She later denounced the 31-minute response time as “inexcusable” and “outrageous.”

But a closer look suggests the responders understood something this would-be mayor does not: priorities.

Keep in mind that 911 calls citywide were up nearly 25 percent in Tuesday’s oppressive heat — and there were 15 other incidents in the immediate vicinity competing for emergency attention. The 911 dispatchers called when intern Yvette Toro fainted were told she was both awake and breathing. Meaning the situation was not life-threatening, a medium-priority call.

When a later 911 caller said Toro appeared to be losing consciousness, the call gained a higher priority. In the meantime she was being treated by someone on the spot with emergency medical training. A private ambulance arrived first, followed by no less than three FDNY ones.

Why is all this important? Look at yesterday’s fire in The Bronx. As The Post reports elsewhere, this has resulted in a fierce dispute between the police and fire departments about whether 911 worked — a dispute likely to land in court.

The point is that New Yorkers deserve two things: to know that the 911 services their lives depend on are working the way they ought — and to have those judgments made when the facts are established, not in the heat of the moment.